The invention relates to the general field of digital modelling.
More particularly, it relates to a method of digitally reconstructing a representative volume element (or RVE) of the microstructure of a composite material, such as for example a composite material having discontinuous long fibers (also known as discontinuous fiber composites (DFCs)) manufactured from “chips” of fibers (e.g. of glass, carbon, etc.) tangled together randomly and pre-impregnated with a thermosetting or thermoplastic resin, also known as a “matrix”.
Such composite materials are particularly well adapted to making parts of complex shape (e.g. including ribs or projections), such as those used in particular in the aviation industry or in numerous other mechanical engineering industries. Such parts are fabricated in known manner from preforms that are cut out from a sheet of material made up of tangled fiber chips, and then assembled in a mold where they are subjected to a thermocompression cycle. The chips may also be inserted in the mold before thermocompression.
The performance of such composite materials depends directly on the tangling of the fiber chips, and suffers from a high degree of variability due to the random nature of the microstructure (or “geometry”) of the composite material at chip scale.
In order to take account of the influence of the microstructure of the composite material (which can vary from one point to another of a part) on the behavior of that part, various multiscale analysis methods are presently in use in industry to predict the mechanical properties at any point of a part as a function of the characteristics of the constituents of the composite material and as a function of their local arrangement. Such multiscale analysis methods make it possible to estimate macroscopic homogeneous properties on the basis of the average response of a representative volume element RVE of the microstructure of the material, i.e. on the basis of a geometrical entity that is statistically representative (i.e. that models) the microstructure of the composite material.
Such methods are also known as “homogenizing” methods. They are in contrast to conventional simulation models in which the behavior of the composite material is assumed in advance (a law governing macroscopic behavior is identified by testing).
In the state of the art, there are several homogenizing methods that make it possible to go from a microscopic scale to a macroscopic scale. The document by P. Kanouté et al. “Multiscale methods for composite”, Arch. Comput. Methods Eng., 2009, 16, pp. 31-75 proposes in particular a digital method using finite elements and having its main steps set out in FIG. 1.
In that method, a representative volume element RVE of the microstructure of the composite material is initially reconstructed (step E10) from characteristic data of the microstructure, e.g. extracted from three-dimensional (3D) images obtained by microscopy or by tomography, or from a predefined mathematical model for generating structures.
The volume as reconstructed in this way is then discretized (step E20) using a conventional finite element method, during which the volume is subdivided into a mesh of finite elements.
Thereafter, the resulting meshed volume is subjected to various predefined loading situations (e.g. shear, traction, etc.) (step E30) and its average response to such loading is estimated, e.g. by performing a finite element calculation (step E40). Homogenized properties of the composite material part are deduced from the response (step E50).
One of the main difficulties with that approach lies in the RVE digital reconstruction step E10 and the discretization step E20, in particular for composite materials such as DFCs (based on discontinuous long fibers) that are fabricated from random tangling of preimpregnated fiber chips and that present a high fiber volume packing ratio. Those steps are based on a priori knowledge of the shape, the geometry, and the positions of the reinforcing fiber elements of the composite material and then on how they pack a predetermined volume in compliance with that shape, that geometry, and those positions.
In the present state of the art, there exist several methods of digitally reconstructing an RVE of composite material.
A first method relies on randomly drawing a plurality of rigid (i.e. undeformable) geometrical shapes in the volume element in order to model the tangled reinforcing fiber elements of the composite material. Each time a new fiber element is drawn, it is not allowed to enter into collision with nor to interpenetrate a fiber element that has already been positioned in the volume, and some minimum distance is imposed between neighboring fiber elements. Although that first method can be implemented quickly, it can nevertheless be understood that the resulting fiber volume packing ratio is very small; typically it does not exceed 40%, which is not representative of reality in the composite materials under consideration.
To mitigate that drawback, one solution consists in performing additional draws of equivalent geometrical shapes but of sizes that are progressively reduced. That solution combined with the preceding solution makes it possible to obtain a packing ratio of about 80%. Nevertheless, such a technique is not pertinent for certain composite materials, and in particular for composite materials having long discontinuous fibers in which the reinforcing fiber chips are of similar sizes relative to one another.
R. Luchoo et al. propose, in a document entitled “Three-dimensional numerical modelling of discontinuous fiber composite architectures”, 18th International Conference on Composite Materials, pp. 356-362, 2011, a second method that is more particularly intended for DFC materials. That second method relies on modelling the tangling of the chips by using chips that are flexible in two-dimensions, corresponding to areas. The shape of the chips is controlled by a set of pilot nodes. Interpenetration is thus generated by using attraction/repulsion algorithms directly on the pilot nodes in order to obtain an optimum solution. The surfaces of the chips are then embedded in a mesh of the volume element. Interpenetration and spacing between the chips within the volume element are therefore not managed strictly.
That second method is thus remote from reality and is not directly usable for predicting failure involving interlaminar stresses at the surfaces of the chips, since the method does not enable them to be evaluated.
A third method is described in a document by Y. Pan et al. entitled “Numerical generation of a random chopped fiber composite RVE and its elastic properties”, Composite Science and Technology 68, pp. 2792-2798, 2008.
That third method relies on randomly drawing same-sized fiber elements in the form of right elliptical cylinders (i.e. strands) that are piled to pack a resin volume element. In that third method, when an intersection is detected between a new fiber element and a fiber element already in position in the volume element, the new fiber element is deformed, and more precisely it is deformed into a single “U-shaped and square-cornered bend” (or “rectangular bend”)(i.e. it is curved with both ends sloping in symmetrical directions) in order to take account of the presence of the element that is already in position. Pan et al. thus take two types of fiber element into consideration for packing the volume element, namely straight fiber elements and bent fiber elements, where bent elements are not representative of reality.
The inclination of a fiber element is initially modelled in two dimensions (2D) in a longitudinal plane of the fiber element, as shown diagrammatically in FIG. 2. In this figure, references A and B designate two initially straight fiber elements characterized by a point of intersection IB situated in the fiber layer FL-INF occupied by the element B. In application of the 2D modelling used, the point IB is moved vertically in translation to a point IA situated at the level of the higher fiber layer FL-SUP. The fiber layers FL-INF and FL-SUP are separated by a layer of resin ML. Two additional points S1 and S2, and respectively S1′ and S2′, are added on either side of the point IA in the fiber layers FL-SUP and FL-INF in order to model the inclination of the fiber element A. The elliptical section of the fiber element is kept unchanged. The points S1, S2 and S1′, S2′ are also selected so as to guarantee some minimum spacing between the fiber elements within the volume. The inclined fiber element A is then reconstructed in three dimensions (3D) by sweeping the space around its longitudinal axis.
In order to facilitate discretization of the representative volume element (RVE) reconstructed using that third method, and more generally in order to facilitate implementing it, large amounts of space are taken into consideration between the piled fiber elements, thereby limiting the packing ratio that can be obtained for the RVE using that method.
There therefore exists a need for a method of digitally reconstructing a representative volume element of a composite material that leads to a high packing ratio and that can be adapted to various types of composite material, and in particular to DFC materials.